On pseudosymmetric para-Kählerian manifolds

Dorota {\L}uczyszyn

Abstract: In the present paper, we consider para-Kählerian manifolds satisfying various curvature conditions of the pseudosymmetric type. Let $(M,J,g)$ be a para-Kählerian manifold. We prove the following theorems: The Ricci-pseudosymmetry of $(M,J,g)$ reduces to the Ricci-semisymmetry. The pseudosymmetry as well as the Bochner-pseudosymmetry and the paraholomorphic projective-pseudosymmetry of the manifold $(M,J,g)$ always reduces to the semisymmetry in dimensions $>4$. The paraholomorphic projective-pseudosymmetry reduces to the pseudosymmetry in dimension $4$. Moreover, we establish new examples of para-Kählerian manifolds being Ricci-semisymmetric (in dimensions $\geqslant6$) as well as pseudosymmetric (in dimension $4$) or Bochner-pseudosymmetric (in dimension $4$). We have given examples of semisymmetric para-Kählerian manifolds in [L1] and [L2].